A theoretical study is presented suggesting possible photoelectron spectroscopy measurement of electron dynamics
on molecules on "attosecond" time scales. In particular, we study analytically the photoionization of
a coherent superposition of molecular electron states by an ultrashort, attosecond X-ray pulse. We show that
the broad photoelectron spectrum inherent in attosecond pulse ionization contains detailed information about
the time-dependent electron wave packet. Such ultrashort pulse photoionization leads to an asymmetry of the
photoelectron momentum distribution and this seems to measure the momentum asymmetry of the initial coherently
prepared electron bound state. We show further that universal molecular Coulomb interference effect occur in ultrashort pulse photoionization and other processes with fast momentum transfers. Thus "attosecond"
photoelectron spectra and their inherent asymmetry allow to monitor and in principle to control electron motion
on the attosecond time scale.